Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3

Average Error: 31.5 → 13.2

Time: 2.6s

Precision: binary64

Cost: 2256

Math

\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

↓

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{if}\;x \cdot x \leq 1.55 \cdot 10^{-267}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 1.9 \cdot 10^{-81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 5.8 \cdot 10^{-16}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 3.7 \cdot 10^{+264}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (<= (* x x) 1.55e-267)
     -1.0
     (if (<= (* x x) 1.9e-81)
       t_1
       (if (<= (* x x) 5.8e-16) -1.0 (if (<= (* x x) 3.7e+264) t_1 1.0))))))

C

double code(double x, double y) {
	return ((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y));
}

↓

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if ((x * x) <= 1.55e-267) {
		tmp = -1.0;
	} else if ((x * x) <= 1.9e-81) {
		tmp = t_1;
	} else if ((x * x) <= 5.8e-16) {
		tmp = -1.0;
	} else if ((x * x) <= 3.7e+264) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * x) - ((y * 4.0d0) * y)) / ((x * x) + ((y * 4.0d0) * y))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = ((x * x) - t_0) / ((x * x) + t_0)
    if ((x * x) <= 1.55d-267) then
        tmp = -1.0d0
    else if ((x * x) <= 1.9d-81) then
        tmp = t_1
    else if ((x * x) <= 5.8d-16) then
        tmp = -1.0d0
    else if ((x * x) <= 3.7d+264) then
        tmp = t_1
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return ((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y));
}

↓

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if ((x * x) <= 1.55e-267) {
		tmp = -1.0;
	} else if ((x * x) <= 1.9e-81) {
		tmp = t_1;
	} else if ((x * x) <= 5.8e-16) {
		tmp = -1.0;
	} else if ((x * x) <= 3.7e+264) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	return ((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y))

↓

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = ((x * x) - t_0) / ((x * x) + t_0)
	tmp = 0
	if (x * x) <= 1.55e-267:
		tmp = -1.0
	elif (x * x) <= 1.9e-81:
		tmp = t_1
	elif (x * x) <= 5.8e-16:
		tmp = -1.0
	elif (x * x) <= 3.7e+264:
		tmp = t_1
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	return Float64(Float64(Float64(x * x) - Float64(Float64(y * 4.0) * y)) / Float64(Float64(x * x) + Float64(Float64(y * 4.0) * y)))
end

↓

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	tmp = 0.0
	if (Float64(x * x) <= 1.55e-267)
		tmp = -1.0;
	elseif (Float64(x * x) <= 1.9e-81)
		tmp = t_1;
	elseif (Float64(x * x) <= 5.8e-16)
		tmp = -1.0;
	elseif (Float64(x * x) <= 3.7e+264)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y));
end

↓

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	tmp = 0.0;
	if ((x * x) <= 1.55e-267)
		tmp = -1.0;
	elseif ((x * x) <= 1.9e-81)
		tmp = t_1;
	elseif ((x * x) <= 5.8e-16)
		tmp = -1.0;
	elseif ((x * x) <= 3.7e+264)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1.55e-267], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 1.9e-81], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 5.8e-16], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 3.7e+264], t$95$1, 1.0]]]]]]

Target

Original	31.5
Target	31.2
Herbie	13.2

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.5500000000000001e-267 or 1.8999999999999999e-81 < (*.f64 x x) < 5.7999999999999996e-16

   1. Initial program 26.6

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 12.7

      \[\leadsto \color{blue}{-1} \]

   if 1.5500000000000001e-267 < (*.f64 x x) < 1.8999999999999999e-81 or 5.7999999999999996e-16 < (*.f64 x x) < 3.6999999999999999e264

   1. Initial program 16.1

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 3.6999999999999999e264 < (*.f64 x x)

   1. Initial program 57.9

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around inf 9.9

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.55 \cdot 10^{-267}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 1.9 \cdot 10^{-81}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 5.8 \cdot 10^{-16}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 3.7 \cdot 10^{+264}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Error	16.7
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+25}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	32.1
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4.0))) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4.0)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))